Recently,it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the sim-ple density wave propagation example ...Recently,it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the sim-ple density wave propagation example for the compressible Euler equations.The issue is related to missing local linear stability,i.e.,the stability of the discretization towards per-turbations added to a stable base flow.This is strongly related to an anti-diffusion mech-anism,that is inherent in entropy-conserving two-point fluxes,which are a key ingredi-ent for the high-order discontinuous Galerkin extension.In this paper,we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations.Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation.We present the full theoretical derivation,analysis,and show corresponding numerical results to underline our findings.In addition,we characterize numerical fluxes for the Euler equations that are entropy-conservative,kinetic-energy-preserving,pressure-equilibrium-preserving,and have a density flux that does not depend on the pressure.The source code to reproduce all numerical experiments presented in this article is available online(https://doi.org/10.5281/zenodo.4054366).展开更多
A new weak boundary procedure for hyperbolic problems is presented.We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique.The new boun...A new weak boundary procedure for hyperbolic problems is presented.We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique.The new boundary procedure is applied near boundaries in an extended domain where data is known.We show how to raise the order of accuracy of the scheme,how to modify the spectrum of the resulting operator and how to construct non-reflecting properties at the boundaries.The new boundary procedure is cheap,easy to implement and suitable for all numerical methods,not only finite difference methods,that employ weak boundary conditions.Numerical results that corroborate the analysis are presented.展开更多
基金Funded by the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)under Germany’s Excellence Strategy EXC 2044-390685587Mathematics Münster:Dynamics-Geometry-Structure.Gregor Gassner is supported by the European Research Council(ERC)under the European Union’s Eights Framework Program Horizon 2020 with the research project Extreme,ERC Grant Agreement No.714487.
文摘Recently,it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the sim-ple density wave propagation example for the compressible Euler equations.The issue is related to missing local linear stability,i.e.,the stability of the discretization towards per-turbations added to a stable base flow.This is strongly related to an anti-diffusion mech-anism,that is inherent in entropy-conserving two-point fluxes,which are a key ingredi-ent for the high-order discontinuous Galerkin extension.In this paper,we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations.Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation.We present the full theoretical derivation,analysis,and show corresponding numerical results to underline our findings.In addition,we characterize numerical fluxes for the Euler equations that are entropy-conservative,kinetic-energy-preserving,pressure-equilibrium-preserving,and have a density flux that does not depend on the pressure.The source code to reproduce all numerical experiments presented in this article is available online(https://doi.org/10.5281/zenodo.4054366).
基金supported by the National Science Foundation under Award No.0948304 and by the Southern California Earthquake Center.SCEC is funded by NSF Cooperative Agreement EAR-0529922 and USGS Cooperative Agreement 07HQAG0008(SCEC contribution number 1806).The work by the last author was carried out within the Swedish e-science Research Centre(SeRC)and supported by the Swedish Research Council(VR).
文摘A new weak boundary procedure for hyperbolic problems is presented.We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique.The new boundary procedure is applied near boundaries in an extended domain where data is known.We show how to raise the order of accuracy of the scheme,how to modify the spectrum of the resulting operator and how to construct non-reflecting properties at the boundaries.The new boundary procedure is cheap,easy to implement and suitable for all numerical methods,not only finite difference methods,that employ weak boundary conditions.Numerical results that corroborate the analysis are presented.
